Let r and m be real numbers so that the sum S_r,m(x) = ∑_{p ≤ x} p^r log^m p diverges as x → ∞. Here p runs over all primes not exceeding x. In this paper, we give an asymptotic formula for each S_r,m(x) as x → ∞. The case where x is the nth prime number is of particular interest. Here we use a method developed by Salvy to give an asymptotic formula for S_r,m(p_n) as n → ∞, which generalizes, for instance, the previously known one for S_1,0(p_n), the sum of the first n prime numbers.